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S. Braun, A. Kluwick, M.S. Cramer:
"The effect of thermal expansion on nonlinear first and second sound in the whole temperature region of He II";
Physics of Fluids, 8 (1996), 9; S. 2513 - 2526.

We consider one-dimensional, weakly nonlinear first and second sound waves in
He II. The first correction to the shock speed is computed for each sound mode.
The expressions obtained are exact with respect to the coefficient of thermal
expansion beta. It is shown that the commonly made assumption of negligible
beta can lead to significant error in the shock speeds for first sound away
from the lambda-line. This contrasts with the calculation of the linear sound
speeds and the shock speed for second sound where the beta=0 approximation
yields accurate results. Near the lambda-line, the exact expressions for both
modes are seen to contain fundamentally different singularities than those found
in the commonly employed beta=0 theory.

http://dx.doi.org/10.1063/1.869034